Effects of Reduced Measurement Independence on Bell-Based Randomness Expansion

With the advent of quantum information, the violation of a Bell inequality is used to witness the absence of an eavesdropper in cryptographic scenarios such as key distribution and randomness expansion. One of the key assumptions of Bell’s theorem is the existence of experimental “free will,” meaning that measurement settings can be chosen at random and independently by each party. The relaxation of this assumption potentially shifts the balance of power towards an eavesdropper. We consider a no-signaling model with reduced “free will” and bound the adversary’s capabilities in the task of randomness expansion.

Figure 1

The maximal CHSH expectation value $S^{\max }(G,P)$ plotted against the free will parameter $P$, for the no-signaling (NS) $G=1$ model (solid line), and the quantum (Q) $G=\frac{1}{2}$ model (dashed line). Region III (unshaded) can be explained by a deterministic $G=1$ model. In regions I (darker gray) and II (lighter gray), the results cannot be deterministic; Eve cannot know the outputs with certainty. Regions II and III together give the set of ($S$, $P$) values that may be attained by a quantum model.

Figure 2

Optimal guessing probability $G(S,P)$ for no-signaling models at different CHSH expectation values, including $S=2$ (local deterministic) and $S=2\sqrt{2}$ (Tsirelson bound). In the cases of general and factorizable distributions, the optimal guessing probabilities approach the vertical dotted lines as $S$ goes to 4.